Integrand size = 29, antiderivative size = 128 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^4 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}+\frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a \tan (c+d x)}{\left (a^2-b^2\right ) d} \]
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Time = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2977, 2702, 327, 213, 2700, 14, 2775, 12, 2739, 632, 210} \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^4 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{3/2}}+\frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 d \left (a^2-b^2\right )}-\frac {b \sec (c+d x)}{a^2 d}+\frac {\tan (c+d x)}{a d}-\frac {\cot (c+d x)}{a d} \]
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Rule 12
Rule 14
Rule 210
Rule 213
Rule 327
Rule 632
Rule 2700
Rule 2702
Rule 2739
Rule 2775
Rule 2977
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b \csc (c+d x) \sec ^2(c+d x)}{a^2}+\frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a}+\frac {b^2 \sec ^2(c+d x)}{a^2 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {\int \csc ^2(c+d x) \sec ^2(c+d x) \, dx}{a}-\frac {b \int \csc (c+d x) \sec ^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {\sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 \left (a^2-b^2\right ) d}-\frac {b^2 \int \frac {b^2}{a+b \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {b \sec (c+d x)}{a^2 d}-\frac {b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 \left (a^2-b^2\right ) d}-\frac {b^4 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 \left (a^2-b^2\right ) d}+\frac {\tan (c+d x)}{a d}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right ) d} \\ & = \frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 \left (a^2-b^2\right ) d}+\frac {\tan (c+d x)}{a d}+\frac {\left (4 b^4\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right ) d} \\ & = -\frac {2 b^4 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}+\frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 \left (a^2-b^2\right ) d}+\frac {\tan (c+d x)}{a d} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.60 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-\frac {4 b^4 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2}}-\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{a}+\frac {2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}-\frac {2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{a}}{2 d} \]
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Time = 0.55 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {2 b^{4} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a -b \right ) \left (a +b \right ) \sqrt {a^{2}-b^{2}}}-\frac {1}{\left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(155\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {2 b^{4} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a -b \right ) \left (a +b \right ) \sqrt {a^{2}-b^{2}}}-\frac {1}{\left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(155\) |
risch | \(-\frac {2 \left (-i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+a b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i a^{2}-i b^{2}-a b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (a^{2}-b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}\) | \(308\) |
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (123) = 246\).
Time = 0.52 (sec) , antiderivative size = 582, normalized size of antiderivative = 4.55 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\left [\frac {\sqrt {-a^{2} + b^{2}} b^{4} \cos \left (d x + c\right ) \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) \sin \left (d x + c\right ) + 2 \, a^{5} - 2 \, a^{3} b^{2} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, {\left (2 \, a^{5} - 3 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right )}, \frac {2 \, \sqrt {a^{2} - b^{2}} b^{4} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, a^{5} - 2 \, a^{3} b^{2} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, {\left (2 \, a^{5} - 3 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\csc ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (123) = 246\).
Time = 0.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.02 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {12 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{4}}{{\left (a^{4} - a^{2} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3} - 3 \, a b^{2}}{{\left (a^{4} - a^{2} b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}}{6 \, d} \]
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Time = 14.48 (sec) , antiderivative size = 778, normalized size of antiderivative = 6.08 \[ \int \frac {\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (a\,b^6\,1{}\mathrm {i}-a^7\,\cos \left (2\,c+2\,d\,x\right )\,2{}\mathrm {i}-a^3\,b^4\,2{}\mathrm {i}+a^5\,b^2\,1{}\mathrm {i}+a\,b^6\,\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}-a^6\,b\,\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}-a^2\,b^5\,\sin \left (c+d\,x\right )\,2{}\mathrm {i}+a^4\,b^3\,\sin \left (c+d\,x\right )\,4{}\mathrm {i}-a^3\,b^4\,\cos \left (2\,c+2\,d\,x\right )\,4{}\mathrm {i}+a^5\,b^2\,\cos \left (2\,c+2\,d\,x\right )\,5{}\mathrm {i}+b^7\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}-a^2\,b^5\,\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+a^4\,b^3\,\sin \left (2\,c+2\,d\,x\right )\,2{}\mathrm {i}-a^6\,b\,\sin \left (c+d\,x\right )\,2{}\mathrm {i}-a^6\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+2\,b^4\,\sin \left (2\,c+2\,d\,x\right )\,\mathrm {atan}\left (\frac {a^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}\,1{}\mathrm {i}+b^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}\,4{}\mathrm {i}-a^2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}\,3{}\mathrm {i}+a\,b^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}\,2{}\mathrm {i}-a^3\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^7+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^6\,b-3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5\,b^2-7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b^3+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^4+9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^5-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^6-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^7}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}-a^2\,b^5\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (2\,c+2\,d\,x\right )\,3{}\mathrm {i}+a^4\,b^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (2\,c+2\,d\,x\right )\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{a^2\,d\,\sin \left (2\,c+2\,d\,x\right )\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )} \]
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